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Answer :
Let's solve this linear programming (LP) problem step by step.
### Problem Statement:
Minimize the objective function [tex]\(4x_1 + x_2\)[/tex]
Subject to the constraints:
1. [tex]\(3x_1 + x_2 = 30\)[/tex]
2. [tex]\(4x_1 + 3x_2 \geq 60\)[/tex]
3. [tex]\(x_1 + 2x_2 \leq 40\)[/tex]
4. [tex]\(x_1, x_2 \geq 0\)[/tex]
### Step 1: Introducing Slack and Artificial Variables
To handle the equality and inequalities, we introduce slack and artificial variables.
1. For the equality constraint [tex]\(3x_1 + x_2 = 30\)[/tex], we introduce an artificial variable [tex]\(x_3\)[/tex]:
[tex]\[
3x_1 + x_2 + x_3 = 30
\][/tex]
2. For the inequality constraint [tex]\(4x_1 + 3x_2 \geq 60\)[/tex], we rewrite it as:
[tex]\[
4x_1 + 3x_2 - x_4 = 60 \quad \text{(introducing artificial variable } x_4)
\][/tex]
3. For the inequality constraint [tex]\(x_1 + 2x_2 \leq 40\)[/tex], we rewrite it as:
[tex]\[
x_1 + 2x_2 + x_6 = 40 \quad \text{(introducing slack variable } x_6)
\][/tex]
### Step 2: Formulating the Objective Function
To account for artificial variables, we modify the objective function using a large penalty approach with [tex]\(M = 100\)[/tex]:
[tex]\[
\text{Minimize } 4x_1 + x_2 + 0x_3 + 100x_4 + 100x_5 + 0x_6
\][/tex]
### Step 3: Linear Programming Formulation
Now, we compile all the constraints and the objective function:
[tex]\[
\begin{aligned}
& \text{Minimize } 4x_1 + x_2 + 100x_4 + 0x_3 + 0x_6 \\
& \text{Subject to:} \\
& 3x_1 + x_2 + x_3 = 30 \\
& 4x_1 + 3x_2 - x_4 = 60 \\
& x_1 + 2x_2 + x_6 = 40 \\
& x_1, x_2, x_3, x_4, x_6 \geq 0
\end{aligned}
\][/tex]
### Step 4: Solving the Linear Program
By solving this LP problem using the Simplex or another suitable method, we obtain the result:
[tex]\[
(x_1, x_2, x_3, x_4, x_5, x_6) = (0.0, 0.0, 30.0, 0.0, 0.0, 0.0)
\][/tex]
### Step 5: Interpretation of the Results
From the solution:
1. [tex]\(x_1 = 0.0\)[/tex]
2. [tex]\(x_2 = 0.0\)[/tex]
3. [tex]\(x_3 = 30.0\)[/tex] (artificial variable for the equality constraint)
4. [tex]\(x_4 = 0.0\)[/tex]
5. [tex]\(x_5 = 0.0\)[/tex] (this was initially introduced as an artificial variable for reformulation)
6. [tex]\(x_6 = 0.0\)[/tex]
The minimal value of the objective function [tex]\(4x_1 + x_2\)[/tex] is [tex]\(0\)[/tex], given the optimal values of [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex].
This means that both [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] are zero, satisfying all constraints. The total cost is zero since both [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] are zero.
### Problem Statement:
Minimize the objective function [tex]\(4x_1 + x_2\)[/tex]
Subject to the constraints:
1. [tex]\(3x_1 + x_2 = 30\)[/tex]
2. [tex]\(4x_1 + 3x_2 \geq 60\)[/tex]
3. [tex]\(x_1 + 2x_2 \leq 40\)[/tex]
4. [tex]\(x_1, x_2 \geq 0\)[/tex]
### Step 1: Introducing Slack and Artificial Variables
To handle the equality and inequalities, we introduce slack and artificial variables.
1. For the equality constraint [tex]\(3x_1 + x_2 = 30\)[/tex], we introduce an artificial variable [tex]\(x_3\)[/tex]:
[tex]\[
3x_1 + x_2 + x_3 = 30
\][/tex]
2. For the inequality constraint [tex]\(4x_1 + 3x_2 \geq 60\)[/tex], we rewrite it as:
[tex]\[
4x_1 + 3x_2 - x_4 = 60 \quad \text{(introducing artificial variable } x_4)
\][/tex]
3. For the inequality constraint [tex]\(x_1 + 2x_2 \leq 40\)[/tex], we rewrite it as:
[tex]\[
x_1 + 2x_2 + x_6 = 40 \quad \text{(introducing slack variable } x_6)
\][/tex]
### Step 2: Formulating the Objective Function
To account for artificial variables, we modify the objective function using a large penalty approach with [tex]\(M = 100\)[/tex]:
[tex]\[
\text{Minimize } 4x_1 + x_2 + 0x_3 + 100x_4 + 100x_5 + 0x_6
\][/tex]
### Step 3: Linear Programming Formulation
Now, we compile all the constraints and the objective function:
[tex]\[
\begin{aligned}
& \text{Minimize } 4x_1 + x_2 + 100x_4 + 0x_3 + 0x_6 \\
& \text{Subject to:} \\
& 3x_1 + x_2 + x_3 = 30 \\
& 4x_1 + 3x_2 - x_4 = 60 \\
& x_1 + 2x_2 + x_6 = 40 \\
& x_1, x_2, x_3, x_4, x_6 \geq 0
\end{aligned}
\][/tex]
### Step 4: Solving the Linear Program
By solving this LP problem using the Simplex or another suitable method, we obtain the result:
[tex]\[
(x_1, x_2, x_3, x_4, x_5, x_6) = (0.0, 0.0, 30.0, 0.0, 0.0, 0.0)
\][/tex]
### Step 5: Interpretation of the Results
From the solution:
1. [tex]\(x_1 = 0.0\)[/tex]
2. [tex]\(x_2 = 0.0\)[/tex]
3. [tex]\(x_3 = 30.0\)[/tex] (artificial variable for the equality constraint)
4. [tex]\(x_4 = 0.0\)[/tex]
5. [tex]\(x_5 = 0.0\)[/tex] (this was initially introduced as an artificial variable for reformulation)
6. [tex]\(x_6 = 0.0\)[/tex]
The minimal value of the objective function [tex]\(4x_1 + x_2\)[/tex] is [tex]\(0\)[/tex], given the optimal values of [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex].
This means that both [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] are zero, satisfying all constraints. The total cost is zero since both [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] are zero.
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